Can a piece of matter be compressed to infinite density? Is there an atomic limit?

Question: If you compress an object like earth more and more you result in an increasing gravitational force and eventually a black hole. How is an object compressed beyond the physical space that the atoms comprise? Is there a minimum size to which an object can compress?

First off, if you compress an object, its gravity won't really change, since its mass hasn't. True, compress it enough, it could form a black hole, but (without working out the numbers) its event horizon would be very small indeed. Granted, things would keep falling towards the "ground". Since density is mass per unit volume, and the density is what becomes infinite, it's the volume that goes to zero, so, yes, the minimum size is volume-less (is that a word?).

I believe the Schwarzschild radius for the mass of the Earth is around 8 inches, or 20 centimeters. So if you compressed the mass of the Earth into a sphere smaller than that radius an event horizon would form, and thereafter you couldn't tell from the outside what further compression happened to e matter of Earth.

Quantum degeneracy pressure, due to the Pauli principle, is finite and can be exceeded by a sufficiently great force. I believe it was Oppenheimer with a coworker who showed that a mass over three times that of the Sun would have sufficient force under self collapse to overcome the degeneracy pressure and shrink to an infinitely dense singularity.

How is an object compressed beyond the physical space that the atoms comprise? Is there a minimum size to which an object can compress?

Remember that atoms are not fundamental particles, they are made up of protons, neutrons and electrons (with the protons and neutrons in turn being made up of quarks), so they are mostly empty space. Their "size" just has to do with the distance of the electrons from the nucleus.

No one is really sure what the size of fundamental particles like electrons and quarks is; string theory says they are really small loops about the size of the Planck length (which is vastly smaller than the size of an atom), while older theories treated them as mathematical points of zero size. Although general relativity says that there is no upper limit on how much you can compress matter, theories of quantum gravity might say that it cannot be compressed beyond the Planck density, which is around one Planck mass per Planck volume (Planck length cubed).

Question: If you compress an object like earth more and more you result in an increasing gravitational force and eventually a black hole. How is an object compressed beyond the physical space that the atoms comprise? Is there a minimum size to which an object can compress?

To amplify on some of the earlier responses.

As you compress normal matter more and more, first it turns into electron degenerate matter - white dwarf star material. The electrons in such electron degenerate matter are not orbiting or associated with any particular nucleus, so you do not have familiar atomic matter at this point.

A good model for such a situation is the "particle in a box" model. You have electrons in the box, you have nuclei in the box, none of the electrons are associated in particular with any nucleus, they just share the same box. If you work out the problem in detail, the main component of pressure in the box is due to the electrons (which cannot share the same quantum state by Pauli's exclusion principle). The nucleii, being more massive, do not contribute as much to the pressure as do the lighter electrons.

After you compress normal matter even more, it turns into neutronium. Essentially the electrons are forced to combine with the protons and form neutrons, and the matter becomes one giant nucleus.

This is the "neutron star" level of density, it is significantly more dense than the electron degenerate matter found in white dwarfs. (In a real neutron star, the pressure varies with depth, so real neutron stars actually have a relative complex structure that varies with depth).

There is some speculation about other possible forms of matter involving "strange" quarks, but I don't know much about the details.

Eventually, though, with the ultimate amount of compression, we expect black holes to form.

However, by far the easiest (and perhaps the only feasible) way to compress matter to such densities is by gravity itself. This is why black holes are expected to form mainly by massive stars running out of fuel.

So the short answer is that atomic structure is destroyed relatively early in the compression process.

First off, if you compress an object, its gravity won't really change, since its mass hasn't.............

This is only partially true in Newtonian physics and untrue in GR.

Firstly, in Newtonian physics, it's not ONLY the mass which determines the strength of the gravitational attraction but also the inverse square of distance to the center of mass. Thus although an object such as the Moon would feel no change if the Earth was compressed, the material of which the Earth itself is composed would.

For example, if the Earth shrank to half its size, the gravitational acceleration at 6000 Km away from the center would still be 9 m/s/s , but on the surface (now 3000 Km from the center) the acceleration would be 36 m/s/s.

Secondly, under GR, pressure as well as mass, contributes to the total "force". So, in this case, distant bodies such as the Moon, as well as the material of the Earth, would feel an increase in gravity as the Earth shrank.
The "force" acting on the Earth would, of course, be somewhat greater than that predicted by Newtonian physics.